Refining the Neutron Star Mass Determination in Six Eclipsing X-ray Pulsar Binaries Meredith L. Rawls Jerome A. Orosz April 26, 2010
Overview • X-ray pulsar and neutron star primer • Introduction to the six systems • How masses have been determined in the past (analytic method) • Our new and improved numerical method using the Eclipsing Light Curve code (ELC) – Why this technique is superior to the analytic method – How ELC works with MCMC or genetic optimizers • Incorporating optical light curves • Results: new values for the neutron star masses
What is an X-ray pulsar? • “Normal” companion star and neutron star orbiting each other • X-rays are produced as matter is pulled away from the companion star toward the neutron star
Why study X-ray pulsars? • Neutron stars are extremely dense collections of matter • Neutron stars in binaries are easy to detect and study • An empirical mass range would enable theorists to better understand NS formation and constrain possible equations of state (EoS) – A “stiff” EoS put upper mass limit ~ 3 M ʘ – A “soft” EoS puts upper mass limit ~ 1.5 M ʘ – Formation theory constrains lower mass limit • Goal of this study: determine the mass of the neutron star in six systems
Meet the six systems • Vela X-1 – Eccentric orbit ( e = 0.09), P = 8.96 days – Pulsar rotates every 283 seconds – Companion star is a B0.5 supergiant • 4U 1538-52 – Eccentric orbit ( e ~ 0.18), P = 3.73 days – Pulsar rotates every 529 Seconds – Companion star is a B0 supergiant • SMC X-1 – Circular orbit, P = 3.89 days – Pulsar rotates every 0.71 seconds – Companion star is a B0 supergiant – Superorbital X-ray cycle observed
Meet the six systems • LMC X-4 – Circular orbit, P = 1.41 days – Pulsar rotates every 13.5 seconds, companion O7 III-V star – Superorbital X-ray cycle observed • Cen X-3 – Circular orbit, P = 2.09 days – Pulsar rotates every 4.84 seconds, companion O6.5 giant • Her X-1 – Circular orbit, P = 1.70 days – Pulsar rotates every 1.24 seconds – Lower mass companion star (~ 2 M ʘ ) with variable spectral type – Extreme X-ray heating, superorbital cycle observed
Mass determination: analytic approach • Component masses in terms of “mass functions” 3 / 2 3 2 K P 1 e 2 X 1 M q opt 3 2 G sin i 3 / 2 2 3 2 K P 1 e 1 opt 1 M X 3 2 G sin i q K M opt – Mass ratio is X q M K opt X
Mass determination: analytic approach • Measure K X and P from X-ray pulse timing – a X sin i is cited in publications: 3 2 2 2 c 1 e a sin i X K X P • Measure K opt from optical spectra
Mass determination: analytic approach • For a spherical companion star, we can relate the eclipse duration, radius, system inclination, and orbital separation 2 1 R a sin i cos e – θ e is an angle that represents half of the eclipse duration • But the companion star is NOT spherical!
Mass determination: analytic approach • Rewrite the radius as a fraction of the “effective Roche lobe radius” R R L 2 2 1 R L a sin i cos Roche lobe filling factor e (if the orbit is eccentric, β • Use an approximation for RL/a is defined at periastron) R L 2 A B log q C log q a – Constants A , B , and C depend on the ratio of the rotational frequency of the optical companion to the orbital frequency of the system, Ω
Mass determination: analytic approach • Given values of P , a X sin i , θ e , K opt , Ω , β (plus e , ω if the orbit is eccentric) we can determine the neutron star mass! – Can estimate Ω from the projected rotational velocity of the companion star, v rot sin i – Must assume some value for β – Expect 0.9 < β < 1 • Can use a Monte Carlo technique to derive the most likely mass (measured input quantities are not known exactly )
Examining the approximations, 1 • Computing R L / a – Shape and size of Roche lobes depend only on the mass ratio q and the parameter Ω R L 2 A B log q C log q a • Compare to result from Eclipsing Light Curve (ELC) code – Defines equipotential surfaces based on the gravitational potential at L1
Examining the approximations, 2 • Computing the X-ray eclipse duration, 2 θ e – Depends on the computation of R L / a 2 2 1 R L a sin i cos e • Compare to result from Eclipsing Light Curve (ELC) code – Uses the Roche lobe shape of the star rather than a spherical approximation
Examining the approximations, 2 β = 1.0 β = 0.9
Examining the approximations, 2 • Difference in eclipse duration can be extreme ( ± 10 ° ) • Directly impacts neutron star mass calculation
Mass determination: numerical method • Parameter space to search • Fix P and a X sin i (known to high accuracy) – Orbital period, P 3 2 2 2 1 sin – Orbital separation, a c e a i X K X – Mass ratio, q P K q opt K 1 X 1 a c a X q K opt – Roche lobe filling factor, β From v rot sin i – Synchronous rotation parameter, Ω – System inclination, i – Eccentric orbit parameters: e , ω
Mass determination: numerical method • We have a six-dimensional parameter space: – ( K opt , β , Ω , i , e , ω ) • Can use ELC to form a model binary system when these six parameters are specified – Values for each parameter are available from previously published works for all six systems • Need a way to choose the BEST model…
Mass determination: numerical method • ELC forms a random set of parameters • “Fitness” of a model is defined by χ 2 (lower = better) (mod) = computed from model (obs) = observed quantity σ ( ) = 1 σ uncertainty • One of two “optimizers” is used to construct new parameter sets • Process is repeated until χ 2 is minimized
Mass determination: numerical method • ELC can use two different optimizers: Monte Carlo Markov Chain or a genetic algorithm • Monte Carlo Markov Chain – Optimizer takes a “random walk” step for each parameter – Given the present state, past and future states are independent – Model with highest fitness is the next starting point • Genetic algorithm – Probability of previous models “breeding” is based on fitness – Random variations (“mutations”) are introduced – Models with highest fitness are allowed to “breed”
Mass determination: numerical method • One parameter we haven’t constrained: the Roche lobe filling factor, β • Can compute models for a range of β – Recall: we expect roughly 0.9 < β < 1 – System inclination i is inversely correlated with β • Preliminary analysis comparing neutron star masses computed numerically vs. analytically …
Mass determination: numerical method • Neutron star mass is highly dependent on the choice of β • Numerical and analytic results can differ in opposite senses to varying degrees Need a way to constrain β … Optical light curves
Optical light curves 3 1 5 • Ellipsoidal variations – Light from companion star changes with orientation 2 4 • Light curve shape depends on: q , i , Ω , β 2 4 Already well determined from X-ray eclipse width and K -velocities 1 5 3 May be constrained with optical light curves!
Optical light curves • Numerical technique with ELC can be expanded to incorporate new observations • Modified “fitness” function: – Set of N observations with observable quantities y i • Similar terms may be added for additional sets of observations (e.g., radial velocity curve)
Optical light curves • From previous literature for four systems Orbital phase
Optical light curves • Systems from previous figure – Vela X-1, SMC X-1, LMC X-4, Cen X-3 – All models include an accretion disk for the best fit • 4U 1538-52 – We obtained new observations – BVI images light curve – High resolution spectra radial velocity curve • Her X-1 – No optical light curves used due to large uncertainty in K opt – Previous literature suggests β ≈ 1
Sample final model: Cen X-3
New observations: 4U 1538-52 • Eccentricity e given as 0.08, 0.18 (sometimes e = 0 is adopted) • Argument of periastron ω given as 244 ° , 220 ° • Obtained BVI images at CTIO – 1.3 m SMARTS telescope with the ANDICAM – 39 images, June – September 2009 • Obtained high resolution spectra at LCO – 6.5 m Clay Magellan telescope with the MIKE spectrograph – 21 images, July – August 2009
New Observations: 4U 1538-52 Optical light curve Radial velocity curve Velocities calculated via cross- correlation of the spectrum with a model B0 star
Final results
Final results
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